NS records exist in two places. The parent zone (for example.com, the .com registry) holds a copy called the delegation, which tells resolvers where to go next. The zone itself also lists its NS records at the apex. The two sets should match; when they drift apart, some resolvers may be sent to servers that no longer host the zone, a condition often called a lame delegation.
You change the delegation at your registrar, not in your DNS hosting panel: the registrar passes the new nameserver names to the registry, which publishes them in the TLD zone. Because TLD zones use long TTLs for delegations, a nameserver change can take a day or two before every resolver follows it.
At least two nameservers on different networks are recommended so that the zone stays resolvable when one of them is unreachable.
Example
example.com. 86400 IN NS ns1.dns-host.example.
example.com. 86400 IN NS ns2.dns-host.example.